where \(L^{(2)}[Q]\)  is the corresponding conservative Lagrangian.
This result does not agree with Eq.205 and Eq.207 of [1]}, as noticed in the previous section. However, we cannot say that these conservative Lagrangians \(L^{(1)}[Q],L^{(2)}[Q]\)  really encodes the conservative dynamics. In fact the In-In formalism has been designed to get the equations of motions of the system and not the conservative Lagrangians. The conservative part is not uniquely defined, and one can always add terms in the action that vanish in the equations of motion, that modify the conservative part.
